The quasiperiodic 1D functions \(V^{\text{d}}_{n}\)  and \(V^{\text{od}}_{n,m}\)  can be constructed, for example, by primitive substitutions [1]}, [2]} to iteratively generate a non-repeating sequence of discrete values. Known examples involving two discrete values \(a\)  and \(b\)  are the Fibonacci (\(a\rightarrow ab\) , \(b\rightarrow a\) ) [3]}, [4]}, [5]}, Thue-Morse (\(a\rightarrow ab\) , \(b\rightarrow ba\) ) [6]} and Period-doubling sequences (\(a\rightarrow ab\) , \(b\rightarrow aa\) ) [6]}, [8]}, see Fig. REF (a) for a 2D example. Alternatively, the method of “cut-and-project” [see Fig. REF (b)] can be employed to generate step-wise (Sturm) functions of the form \(V_{\rm Sturm}(x)=2( \lfloor (x+2)/\tau \rfloor - \lfloor (x+1)/\tau \rfloor ) - 1 = \pm 1\) , where \(\tau \)  is an irrational number, \(\lfloor x \rfloor \)  is the floor function, and the potentials in Eq. (REF ) sample the continuous potential at integer values \(V(n)\) , thus, enforcing the incommensuration between the periodic lattice and the potential modulation lengthscales [1]}, [10]}. When \(\tau = (1 + \sqrt{5})/2\)  is
the golden ratio, this method also yields the Fibonacci sequence. As another example, quasiperiodic sequences with infinitely-many values can be obtained by an incommensurate sampling of a periodic function, e.g., \(f(n/\tau +\phi )\) , which is periodic under a scan of \(\phi \)  from 0 to \(2\pi \) , see Fig. REF (c). A useful example for our discussion is the interpolating Aubry-André-Fibonacci (IAAF) function [10]}, [12]}, [13]}
\( V_{\text{IAAF}}(\beta ,\tau ,x, \phi )= -\frac{\tanh {\left[ \beta (\cos {(2\pi x/\tau + \phi )} -\cos {(\pi /\tau )}) \right]}}{\tanh \beta }\,,\) 
